Nonlinear_equations.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8pEm4kxLqhN7"
      },
      "source": [
        "Make sure you remove `raise NotImplementedError()` and fill in any place that says `# YOUR CODE HERE`, as well as your `NAME`, `ID`, and `SECTION` below:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "N6FXZz_eqhN9"
      },
      "outputs": [],
      "source": [
        "NAME = \"Mashrur Safir Shabab\"\n",
        "ID = \"20241037\"\n",
        "SECTION = \"07\""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "D1WB3NXNqhN-"
      },
      "source": [
        "---"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "4323c48f5d5276b38804080cab43e0d5",
          "grade": false,
          "grade_id": "title",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "8BwuBD__qhN-"
      },
      "source": [
        "Part 1: Polynomial Root Finding Using Bisection Method\n",
        "---\n",
        "\n",
        "### `pandas` Dataframe:\n",
        "Before we start we will know a little about `pandas` dataframe. `pandas` is a python library. Dataframes are classes that are used to store complex data. You can initialize one as a python dictionary. Let's take a look. "
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "adc6b91c35993f4ecedcfd53d807a206",
          "grade": false,
          "grade_id": "pd",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "F1iPegcMqhN-",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
        },
        "outputId": "01420429-3199-489b-d3b5-c85a2bff2ad9"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "   x     y\n",
              "0  0   1.0\n",
              "1  1   5.2\n",
              "2  2   3.9\n",
              "3  3   8.4\n",
              "4  4  14.6\n",
              "5  5  12.3\n",
              "6  6   8.9\n",
              "7  7  10.2"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-b252006e-b8d3-42c1-b34c-cf2ff49e9a0a\">\n",
              "    <div class=\"colab-df-container\">\n",
              "      <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>x</th>\n",
              "      <th>y</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>0</td>\n",
              "      <td>1.0</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>1</td>\n",
              "      <td>5.2</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>2</td>\n",
              "      <td>3.9</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>3</td>\n",
              "      <td>8.4</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>4</td>\n",
              "      <td>14.6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>5</td>\n",
              "      <td>12.3</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>6</td>\n",
              "      <td>8.9</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>7</td>\n",
              "      <td>10.2</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-b252006e-b8d3-42c1-b34c-cf2ff49e9a0a')\"\n",
              "              title=\"Convert this dataframe to an interactive table.\"\n",
              "              style=\"display:none;\">\n",
              "        \n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "       width=\"24px\">\n",
              "    <path d=\"M0 0h24v24H0V0z\" fill=\"none\"/>\n",
              "    <path d=\"M18.56 5.44l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94zm-11 1L8.5 8.5l.94-2.06 2.06-.94-2.06-.94L8.5 2.5l-.94 2.06-2.06.94zm10 10l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94z\"/><path d=\"M17.41 7.96l-1.37-1.37c-.4-.4-.92-.59-1.43-.59-.52 0-1.04.2-1.43.59L10.3 9.45l-7.72 7.72c-.78.78-.78 2.05 0 2.83L4 21.41c.39.39.9.59 1.41.59.51 0 1.02-.2 1.41-.59l7.78-7.78 2.81-2.81c.8-.78.8-2.07 0-2.86zM5.41 20L4 18.59l7.72-7.72 1.47 1.35L5.41 20z\"/>\n",
              "  </svg>\n",
              "      </button>\n",
              "      \n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      flex-wrap:wrap;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "      <script>\n",
              "        const buttonEl =\n",
              "          document.querySelector('#df-b252006e-b8d3-42c1-b34c-cf2ff49e9a0a button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-b252006e-b8d3-42c1-b34c-cf2ff49e9a0a');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
              "          const docLink = document.createElement('div');\n",
              "          docLink.innerHTML = docLinkHtml;\n",
              "          element.appendChild(docLink);\n",
              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ],
      "source": [
        "import pandas as pd\n",
        "\n",
        "x = [0, 1, 2, 3, 4, 5, 6, 7]\n",
        "y = [1.0, 5.2, 3.9, 8.4, 14.6, 12.3, 8.9, 10.2]\n",
        "\n",
        "dictionary = {\n",
        "    \"x\": x,\n",
        "    \"y\": y,\n",
        "}\n",
        "df = pd.DataFrame(dictionary)\n",
        "df.head(8)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "e8de09a00f2f32f8cfbb6b7823ebb992",
          "grade": false,
          "grade_id": "np_into",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "o1qgr3HYqhN_"
      },
      "source": [
        "We can use it to print data in a tabular format. We can even use more numpy arrays with it."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "b7b72bb172e2e46389ca4f41fd18628d",
          "grade": false,
          "grade_id": "np",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "gT5eXzSxqhN_",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 520
        },
        "outputId": "504e29d9-7759-4a8e-a4de-06c7aac0548a"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "           x    sin(x)\n",
              "0   0.000000  0.000000\n",
              "1   0.112200  0.111964\n",
              "2   0.224399  0.222521\n",
              "3   0.336599  0.330279\n",
              "4   0.448799  0.433884\n",
              "5   0.560999  0.532032\n",
              "6   0.673198  0.623490\n",
              "7   0.785398  0.707107\n",
              "8   0.897598  0.781831\n",
              "9   1.009798  0.846724\n",
              "10  1.121997  0.900969\n",
              "11  1.234197  0.943883\n",
              "12  1.346397  0.974928\n",
              "13  1.458597  0.993712\n",
              "14  1.570796  1.000000"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-9ac70fb8-8ace-4e26-b2e8-0d84a2e561cc\">\n",
              "    <div class=\"colab-df-container\">\n",
              "      <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>x</th>\n",
              "      <th>sin(x)</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>0.000000</td>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>0.112200</td>\n",
              "      <td>0.111964</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>0.224399</td>\n",
              "      <td>0.222521</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>0.336599</td>\n",
              "      <td>0.330279</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>0.448799</td>\n",
              "      <td>0.433884</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>0.560999</td>\n",
              "      <td>0.532032</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>0.673198</td>\n",
              "      <td>0.623490</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>0.785398</td>\n",
              "      <td>0.707107</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>0.897598</td>\n",
              "      <td>0.781831</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>1.009798</td>\n",
              "      <td>0.846724</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>10</th>\n",
              "      <td>1.121997</td>\n",
              "      <td>0.900969</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>11</th>\n",
              "      <td>1.234197</td>\n",
              "      <td>0.943883</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>12</th>\n",
              "      <td>1.346397</td>\n",
              "      <td>0.974928</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>13</th>\n",
              "      <td>1.458597</td>\n",
              "      <td>0.993712</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>14</th>\n",
              "      <td>1.570796</td>\n",
              "      <td>1.000000</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "      <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-9ac70fb8-8ace-4e26-b2e8-0d84a2e561cc')\"\n",
              "              title=\"Convert this dataframe to an interactive table.\"\n",
              "              style=\"display:none;\">\n",
              "        \n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "       width=\"24px\">\n",
              "    <path d=\"M0 0h24v24H0V0z\" fill=\"none\"/>\n",
              "    <path d=\"M18.56 5.44l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94zm-11 1L8.5 8.5l.94-2.06 2.06-.94-2.06-.94L8.5 2.5l-.94 2.06-2.06.94zm10 10l.94 2.06.94-2.06 2.06-.94-2.06-.94-.94-2.06-.94 2.06-2.06.94z\"/><path d=\"M17.41 7.96l-1.37-1.37c-.4-.4-.92-.59-1.43-.59-.52 0-1.04.2-1.43.59L10.3 9.45l-7.72 7.72c-.78.78-.78 2.05 0 2.83L4 21.41c.39.39.9.59 1.41.59.51 0 1.02-.2 1.41-.59l7.78-7.78 2.81-2.81c.8-.78.8-2.07 0-2.86zM5.41 20L4 18.59l7.72-7.72 1.47 1.35L5.41 20z\"/>\n",
              "  </svg>\n",
              "      </button>\n",
              "      \n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      flex-wrap:wrap;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "      <script>\n",
              "        const buttonEl =\n",
              "          document.querySelector('#df-9ac70fb8-8ace-4e26-b2e8-0d84a2e561cc button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-9ac70fb8-8ace-4e26-b2e8-0d84a2e561cc');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
              "          const docLink = document.createElement('div');\n",
              "          docLink.innerHTML = docLinkHtml;\n",
              "          element.appendChild(docLink);\n",
              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 11
        }
      ],
      "source": [
        "import numpy as np\n",
        "\n",
        "x = np.linspace(0, np.pi/2, 15)\n",
        "y = np.sin(x)\n",
        "\n",
        "dictionary = {\n",
        "    \"x\": x,\n",
        "    \"sin(x)\": y,\n",
        "}\n",
        "df = pd.DataFrame(dictionary)\n",
        "df.head(15)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "2e1c412437306d2e9b0078176867e80c",
          "grade": false,
          "grade_id": "root_intro",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "2LJFoJfEqhN_"
      },
      "source": [
        "### Root Finding:\n",
        "Let $f(x)$ be a function of $x$. if for some $x=\\alpha, f(x) = f(\\alpha) = 0$, we say $\\alpha$ is a root of function $x$.\n",
        "\n",
        "Let, \n",
        "$$f(x) = x^5 + 2.5x^4 - 2x^3 -6x^2 + 0.5x + 2\\tag{6.1}$$\n",
        "\n",
        "The graph of $f(x)$ looks like this."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "023d32066c1625932a80604228bac0d6",
          "grade": false,
          "grade_id": "example",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "iiIEjH97qhN_",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 282
        },
        "outputId": "5a89b2b7-5803-44d5-e4d0-9e8bb50c5aa2"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[-2.22027583 -1.61803399 -0.65499178  0.61803399  1.37526761]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "from numpy.polynomial import Polynomial\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "f = Polynomial([2.0, 0.5, -6.0, -2.0, 2.5, 1.0])\n",
        "x = np.linspace(-2.5, 1.5, 100)\n",
        "y = f(x)\n",
        "dictionary = {\n",
        "    'x': x,\n",
        "    'y': y\n",
        "}\n",
        "plt.axhline(y=0, color='r')\n",
        "plt.plot(x, y,'y')\n",
        "plt.plot(f.roots(), [0.0]*5, 'go')\n",
        "print(f.roots())"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "a77a2920b884f394a02829392b097131",
          "grade": false,
          "grade_id": "bisection_intro",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "ZDBo-CT3qhOA"
      },
      "source": [
        "Roots are the points where the graph intersects with the $X$-axis.\n",
        "\n",
        "### Bisection\n",
        "\n",
        "One way to find out root's are to use bisection method. Here is the strategy, if $\\alpha$ is a root between and interval $[a, b]$ then graph will cross the $X$-axis at $\\alpha$. So, $sign( f(\\alpha-h) ) = - sign( f(\\alpha+h) )$, for small value of $h$. \n",
        "\n",
        "So, we can work our way up towards the root by taking average of $a$ and $b$, as long as the signs are different.\n",
        "\n",
        "we will start with $a_0$ and $b_0$, such that, $f(a_0) f(b_0) < 0$.\n",
        "Then we iterate as this,\n",
        "\\begin{align}\n",
        "m_k &= \\frac{a_k + b_k}{2} \\\\\n",
        "\\text{if, } f(a_k) f(m_k) < 0, \\text{ then, } a_{k+1} &= a_k \\text{ and } b_{k+1} = m_k\\\\\n",
        "\\text{else, } a_{k+1} &= m_k \\text{ and, } b_{k+1} = b_k\n",
        "\\end{align}\n",
        "\n",
        "We keep iterating until we find the root with sufficient precision. We usually use a formula like this,\n",
        "$$\n",
        "\\frac{|m_{k+1} - m_k|}{|m_{k+1}|} \\leq \\epsilon \\\\  \n",
        "$$\n",
        "Where, $\\epsilon$ is a very small value, like $\\epsilon < 10^{-6}$\n",
        "\n",
        "### Complete the code below\n",
        "Complete the code below to iterate and solve for a root of the following equation, between the interval, $[-0.5, 1.3]$:\n",
        "\\begin{aligned}\n",
        "    f(x) = 2 + 0.5x - 6x^2 - 2x^3 + 2.5x^4 + x^5.\n",
        "\\end{aligned}"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "250514194ec182b83282f90c837cb559",
          "grade": false,
          "grade_id": "ans",
          "locked": false,
          "schema_version": 3,
          "solution": true,
          "task": false
        },
        "id": "vcMvvqN4qhOA"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "from numpy.polynomial import Polynomial\n",
        "\n",
        "# The polynomial and the range is defined for you\n",
        "f = Polynomial([2.0, 0.5, -6.0, -2.0, 2.5, 1.0])\n",
        "a = -0.5\n",
        "b = 1.3\n",
        "m = (a + b) / 2\n",
        "e = 1e-6\n",
        "\n",
        "root = 0.0    # You need to update this value\n",
        "\n",
        "# Populate the following lists in each iteration\n",
        "list_a = []\n",
        "list_b = []\n",
        "list_m = []\n",
        "list_f = []\n",
        "\n",
        "# YOUR CODE HERE\n",
        "found = False\n",
        "while not found:\n",
        "  list_a.append(a)\n",
        "  list_b.append(b)\n",
        "  list_m.append(m)\n",
        "  list_f.append(f(m))\n",
        "\n",
        "  if f(a)*f(m)<0:\n",
        "    a=a\n",
        "    b=m\n",
        "    m = (a+b)/2\n",
        "\n",
        "  elif f(a)*f(m)>0:\n",
        "    a=m\n",
        "    b=b\n",
        "    m = (a+b)/2\n",
        "\n",
        "  elif f(m) == 0:\n",
        "    root = m\n",
        "    found = True\n",
        "\n",
        "  if abs(m-list_m[-1]) / abs(m) <= e:\n",
        "    root = m\n",
        "    found = True"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "62d14c64cdf4fb8a8d96afa669625d11",
          "grade": true,
          "grade_id": "test_0",
          "locked": true,
          "points": 10,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "n80r3n_4qhOB"
      },
      "outputs": [],
      "source": [
        "xs = np.linspace(-2.5, 1.5, 100)\n",
        "ys = f(xs)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "d605d35c8e5cb989a20d59e7bf1dca54",
          "grade": true,
          "grade_id": "test",
          "locked": true,
          "points": 10,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "trVJZs-dqhOB",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 647
        },
        "outputId": "eac5ceb0-5931-4337-ad5d-00d9c3a79021"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "           a         b         m          f(m)\n",
            "0  -0.500000  1.300000  0.400000  1.186240e+00\n",
            "1   0.400000  1.300000  0.850000 -1.389529e+00\n",
            "2   0.400000  0.850000  0.625000 -4.269409e-02\n",
            "3   0.400000  0.625000  0.512500  6.189171e-01\n",
            "4   0.512500  0.625000  0.568750  2.966661e-01\n",
            "5   0.568750  0.625000  0.596875  1.286523e-01\n",
            "6   0.596875  0.625000  0.610938  4.333223e-02\n",
            "7   0.610938  0.625000  0.617969  3.991260e-04\n",
            "8   0.617969  0.625000  0.621484 -2.112851e-02\n",
            "9   0.617969  0.621484  0.619727 -1.035982e-02\n",
            "10  0.617969  0.619727  0.618848 -4.979112e-03\n",
            "11  0.617969  0.618848  0.618408 -2.289682e-03\n",
            "12  0.617969  0.618408  0.618188 -9.452003e-04\n",
            "13  0.617969  0.618188  0.618079 -2.730176e-04\n",
            "14  0.617969  0.618079  0.618024  6.305907e-05\n",
            "15  0.618024  0.618079  0.618051 -1.049781e-04\n",
            "16  0.618024  0.618051  0.618037 -2.095919e-05\n",
            "17  0.618024  0.618037  0.618031  2.105002e-05\n",
            "18  0.618031  0.618037  0.618034  4.543651e-08\n",
            "19  0.618034  0.618037  0.618036 -1.045687e-05\n",
            "20  0.618034  0.618036  0.618035 -5.205715e-06\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "plt.axhline(y=0, color='r')\n",
        "plt.plot(xs, ys,'y')\n",
        "plt.plot(root, f(root), 'go')\n",
        "\n",
        "print(pd.DataFrame({'a':list_a, 'b':list_b, 'm':list_m, 'f(m)':list_f}))\n",
        "\n",
        "assert \"{:.3f}\".format(root) == \"0.618\""
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "M5kTaH-6qhOC"
      },
      "outputs": [],
      "source": [
        ""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "29cb42a7b7b359ce0df31faaa15748c5",
          "grade": false,
          "grade_id": "cell-3123123123",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "UZSMhHrPJr8w"
      },
      "source": [
        "# Part 2: Fixed Point Iteration\n",
        "\n",
        "### Fixed point:\n",
        "A number $\\xi$ is called a **fixed point** to function $g(x)$ if $g(\\xi) = \\xi$. Using fixed points are a nice strategy to find roots of an equation. In this method if we are trying to find a root of $f(x) = 0$, we try to write the function in the form, $x = g(x)$. That is,\n",
        "$$\n",
        "f(x) = x - g(x) = 0\n",
        "$$\n",
        "\n",
        "So, if $\\xi$ is a fixed point of $g(x)$ it would also be a root of $f(x)=0$, because,\n",
        "$$\n",
        "f(\\xi) = \\xi - g(\\xi) = \\xi - \\xi = 0\n",
        "$$\n",
        "\n",
        "We can find a suitable $g(x)$ in any number of ways. Not all of them would converge; whereas, some would converge very fast. For example, consider $Eq. 6.1$.\n",
        "\n",
        "\\begin{align}\n",
        "&         &    f(x) &=x^5 + 2.5x^4 - 2x^3 -6x^2 + x + 2 \\\\\n",
        "&\\implies &x - g(x) &=x^5 + 2.5x^4 - 2x^3 -6x^2 + x + 2 \\\\\n",
        "&\\implies &    g(x) &=-x^5 - 2.5x^4 + 2x^3 + 6x^2 - 2 \\tag{6.2}\\\\\n",
        "\\end{align}\n",
        "\n",
        "again, \n",
        "$$\n",
        "f(x) = x^5 + 2.5x^4 - 2x^3 -6x^2 + x + 2 = 0\\\\\n",
        "$$\n",
        "\\begin{align}\n",
        "&\\implies &6x^2 &= x^5 + 2.5x^4 - 2x^3 + x + 2 \\\\\n",
        "&\\implies &x^2  &= \\frac{1}{6}(x^5 + 2.5x^4 - 2x^3 + x + 2)\\\\\n",
        "&\\implies &x    &= \\sqrt{\\frac{1}{6}(x^5 + 2.5x^4 - 2x^3 + x + 2)}\\\\\n",
        "&\\implies &g(x) &= \\sqrt{\\frac{1}{6}(x^5 + 2.5x^4 - 2x^3 + x + 2)}\\tag{6.3}\\\\\n",
        "\\end{align}\n",
        "\n",
        "Similarly,\n",
        "\\begin{align}\n",
        "&         &2.5x^4 &= -x^5 + 2x^3 + 6x^2 - x - 2 \\\\\n",
        "&\\implies &x^4  &= \\frac{1}{2.5}(-x^5 + 2x^3 + 6x^2 - x - 2)\\\\\n",
        "&\\implies &x    &= \\sqrt[\\leftroot{-1}\\uproot{2}\\scriptstyle 4]{\\frac{1}{2.5}(-x^5 + 2x^3 + 6x^2 - x - 2)}\\\\\n",
        "&\\implies &g(x) &= \\sqrt[\\leftroot{-1}\\uproot{2}\\scriptstyle 4]{\\frac{1}{2.5}(-x^5 + 2x^3 + 6x^2 - x - 2)}\\tag{6.4}\\\\\n",
        "\\end{align}\n",
        "\n",
        "### B. Complete the code below\n",
        "For this example we will use a couple of $g(x)$ function to find out which one converges faster."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "eea775d5231da9918220a769bc5d5d64",
          "grade": false,
          "grade_id": "cell-123123123",
          "locked": false,
          "schema_version": 3,
          "solution": true,
          "task": false
        },
        "id": "7aY87ZtEJr8w"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "from numpy.polynomial import Polynomial\n",
        "\n",
        "f = Polynomial([2.0, 1.0, -6.0, -2.0, 2.5, 1.0])\n",
        "g1 = Polynomial([-2.0, 0.0, 6.0, 2.0, -2.5, -1.0])\n",
        "\n",
        "def g2(x):\n",
        "    p = Polynomial([2.0, 1.0, 0.0, -2.0, 2.5, 1.0])\n",
        "    return np.sqrt(p(x)/6)\n",
        "\n",
        "def g3(x):\n",
        "    p = Polynomial([-2.0, -1.0, 6.0, 2.0, 0.0, -1.0])\n",
        "    return np.power(p(x)/2.5, 1.0/4.0)\n",
        "\n",
        "a1 = 0.80\n",
        "g1_a = []\n",
        "\n",
        "a2 = 0.80\n",
        "g2_a = []\n",
        "\n",
        "a3 = 0.80\n",
        "g3_a = []\n",
        "\n",
        "# YOUR CODE HERE\n",
        "g1_a.append(a1)\n",
        "g2_a.append(a2)\n",
        "g3_a.append(a3)\n",
        "\n",
        "for i in range(20):\n",
        "  x1 = g1(g1_a[i])\n",
        "  g1_a.append(x1)\n",
        "\n",
        "  x2 = g2(g2_a[i])\n",
        "  g2_a.append(x2)\n",
        " \n",
        "  x3 = g3(g3_a[i])\n",
        "  g3_a.append(x3)\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "6d9d325340af410ea2c95de0b2bbb1bc",
          "grade": true,
          "grade_id": "cell-48808fbb6733431f",
          "locked": true,
          "points": 15,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "oip1LLFPJr8y",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 682
        },
        "outputId": "fece87f8-4b78-4a0c-e3a3-1678dafabe15"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/ipykernel_launcher.py:15: RuntimeWarning: invalid value encountered in power\n",
            "  from ipykernel import kernelapp as app\n"
          ]
        },
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "       g1(x)     g2(x)    g3(x))\n",
            "0   0.800000  0.800000  0.800000\n",
            "1   1.512320  0.721997  0.912898\n",
            "2  -2.347605  0.688573  1.044444\n",
            "3   0.562201  0.677301  1.160638\n",
            "4  -0.054108  0.673860  1.241682\n",
            "5  -1.982772  0.672844  1.288278\n",
            "6  -1.995936  0.672547  1.311548\n",
            "7  -1.999770  0.672460  1.322205\n",
            "8  -1.999999  0.672435  1.326870\n",
            "9  -2.000000  0.672427  1.328868\n",
            "10 -2.000000  0.672425  1.329716\n",
            "11 -2.000000  0.672425  1.330075\n",
            "12 -2.000000  0.672424  1.330226\n",
            "13 -2.000000  0.672424  1.330290\n",
            "14 -2.000000  0.672424  1.330317\n",
            "15 -2.000000  0.672424  1.330328\n",
            "16 -2.000000  0.672424  1.330333\n",
            "17 -2.000000  0.672424  1.330335\n",
            "18 -2.000000  0.672424  1.330336\n",
            "19 -2.000000  0.672424  1.330336\n",
            "20 -2.000000  0.672424  1.330336\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "xs = np.linspace(-2.3, 1.5, 100)\n",
        "ys = f(xs)\n",
        "dictionary = {\n",
        "    'x': xs,\n",
        "    'y': ys\n",
        "}\n",
        "plt.axhline(y=0, color='r')\n",
        "plt.plot(xs, f(xs), label='f(x)', color='y')\n",
        "plt.plot(xs, g1(xs), label='g1(x)', color='g')\n",
        "plt.plot(xs, g2(xs), label='g2(x)', color ='b')\n",
        "plt.plot(xs, g3(xs), label='g3(x)', color = 'm' )\n",
        "plt.legend()\n",
        " \n",
        "if len(g1_a) > 0:\n",
        "    root = np.array([g1_a[len(g1_a)-1], g2_a[len(g2_a)-1], g3_a[len(g3_a)-1]])\n",
        "    plt.plot(root, f(root), 'ko')\n",
        " \n",
        "print(pd.DataFrame({'g1(x)':g1_a, 'g2(x)':g2_a, 'g3(x))':g3_a,}))\n",
        " \n",
        "# Test case:\n",
        "np.testing.assert_array_almost_equal(root, [-2, 0.67242436, 1.33033625])"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.4"
    },
    "colab": {
      "name": "20241037_MashrurSafirShabab_07_Nonlinear_equations.ipynb",
      "provenance": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}